Anderson’s double complex and gamma monomials for rational function fields

Abstract

We investigate algebraic Γ \Gamma -monomials of Thakur’s positive characteristic Γ \Gamma -function, by using Anderson and Das’ double complex method of computing the sign cohomology of the universal ordinary distribution. We prove that the Γ \Gamma -monomial associated to an element of the second sign cohomology of the universal ordinary distribution of F q ( T ) \mathbb {F}_{q}(T) generates a Kummer extension of some Carlitz cyclotomic function field, which is also a Galois extension of the base field F q ( T ) \mathbb {F}_{q}(T) . These results are characteristic- p p analogues of those of Deligne on classical Γ \Gamma -monomials, proofs of which were given by Das using the double complex method. In this paper, we also obtain some results on e e -monomials of Carlitz’s exponential function.